COMEDK 2025 Morning Solved Paper

Concept:The expectation of Vasant's gain is the average net gain over all equally likely outcomes of four coin tosses, calculated as the sum of each outcome's gain multiplied by its probability.Explanation:There are $2^4 = 16$ equally likely sequences of heads (H) and tails (T).Let the net gain be the money Vasant receives minus his stake of ₹1.Case 1 — Four heads:Only one sequence: HHHH.Gain = +3 (he gets ₹3 from Jothi plus his stake back).Probability = $\frac{1}{16}$.Case 2 — Exactly three consecutive heads:Sequences: HHHT, THHH. Total 2.Gain = +2.Probability = $\frac{2}{16}$.Case 3 — Exactly two consecutive heads:Sequences: HHTT, THHT, TTHH. Total 3.Gain = +1.Probability = $\frac{3}{16}$.Case 4 — All other outcomes (10 sequences):Gain = –1 (Jothi takes the stake).Probability = $\frac{10}{16}$.The expectation $E[X]$ is:$$E[X] = \frac{1}{16} \times 3 + \frac{2}{16} \times 2 + \frac{3}{16} \times 1 + \frac{10}{16} \times (-1).$$Simplify: $\frac{3 + 4 + 3 - 10}{16} = \frac{0}{16} = 0$.Thus, Vasant's expected gain is zero.Answer:$0$